Determinantal representations for the
J transformation |
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Authors: | Herbert HH Homeier |
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Institution: | Institut für Physikalische und Theoretische Chemie,
Universit?t Regensburg,
D-93040 Regensburg, Germany
e-mail: na.hhomeier@na-net.ornl.gov, DE
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Abstract: | Summary.
The iterative J transformation Homeier, H. H. H. (1993):
Some applications of nonlinear convergence accelerators. Int.
J. Quantum Chem. 45, 545-562] is of similar generality
as the well-known E algorithm Brezinski, C. (1980): A general
extrapolation algorithm. Numer. Math. 35, 175-180.
Havie, T. (1979): Generalized Neville type extrapolation
schemes. BIT 19, 204-213]. The properties of the J
transformation were studied recently in two companion papers
Homeier, H. H. H. (1994a): A hierarchically consistent,
iterative sequence transformation. Numer. Algo. 8, 47-81.
Homeier, H. H. H. (1994b): Analytical and numerical studies
of the convergence behavior of the J transformation. J.
Comput. Appl. Math., to appear]. In the present contribution,
explicit determinantal representations for this sequence
transformation are derived. The relation to the Brezinski-Walz
theory Brezinski, C., Walz, G. (1991): Sequences of
transformations and triangular recursion schemes, with
applications in numerical analysis. J. Comput. Appl. Math.
34, 361-383] is discussed. Overholt's process Overholt,
K. J. (1965): Extended Aitken acceleration. BIT 5,
122-132] is shown to be a special case of the J
transformation. Consequently, explicit determinantal representations
of Overholt's process are derived which do not depend
on lower order transforms. Also, families of sequences are given
for which Overholt's process is exact.
As a numerical example, the Euler series is
summed using the J transformation. The results indicate that the J
transformation is a very powerful numerical tool.
Received May 24, 1994 /
Revised version received November 11, 1994 |
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Keywords: | Mathematics Subject Classification (1991):65B05 65B10 |
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